This invention relates to methods for performing multivariate spectral analyses. More particularly, this invention relates to such methods which combine various features of both classical least squares analysis and the more modern inverse analysis methods such as partial least squares and principle components regression. The method may further include improved prediction ability enabled by adding spectral shapes (or selected spectral intensity data) for certain chemicals or other factors that affect the spectral response that may not have been present when spectral calibration data were obtained.
Classical least squares (CLS) quantitative multivariate calibration methods are based on an explicit or hard physical model (e.g., Beer""s law), (see D. M. Haaland, xe2x80x9cMultivariate Calibration Methods Applied to Quantitative FT-IR Analyses,xe2x80x9d Chapter 8 in Practical Fourier Transform Infrared Spectroscopy, J. R. Ferraro and K. Krishnan, Editors, Academic Press, New York, pp. 396-468, (1989)). During calibration, the CLS method has the advantage that least-squares estimates of the pure-component spectra are obtained from mixture samples. Therefore, significant qualitative spectral information can be obtained from the CLS method about the pure-component spectra as they exist in the calibration mixtures. In addition, the method is readily understood, simple to apply, and when the model is valid, CLS requires fewer calibration samples than the popular inverse-based partial least squares (PLS) and principal component regression (PCR) factor analysis multivariate methods, (see D. M. Haaland and E. V. Thomas, xe2x80x9cPartial Least-Squares Methods for Spectral Analyses 1: Relation to Other Multivariate Calibration Methods and the Extraction of Qualitative Information,xe2x80x9d Analytical Chemistry 60, 1193-1202 (1988)). However, CLS is more restrictive than inverse methods such as PLS and PCR since CLS methods require that information be known about all spectral sources of variation in the samples (i.e., component concentrations and/or spectral shapes must be known, estimated, or derived). Inverse multivariate methods such as PLS and PCR can empirically model interferences and can approximate nonlinear behavior though their inverse soft-modeling approach.
A method for estimating the quantity of at least one known constituent or property in a sample comprising first forming a classical least squares calibration model to estimate the responses of individual pure components of at least one of the constituents or parameters affecting the optical response of the sample and employing a cross validation of the samples in the calibration data set, then measuring the response of the mixture to the stimulus at a plurality of wavelengths to form a prediction data set, then estimating the quantity of one of the known constituents or parameters affecting the calibration data set by a classical least squares analysis of the prediction data set wherein such analysis produces residual errors, and then passing the residual errors to a partial least squares, principal components regression, or other inverse algorithm to provide an improved estimate of the quantity of the one known constituent or parameter affecting the sample. The estimation can be repeated for more of the known constituents in the calibration data set by repeating the last two steps for the other constituents or parameters. Overfitting of the prediction data set by the factor analysis algorithm can be minimized by using only factors derived from each step of the cross validation that are most effective in identifying the constituent or parameter. Also, the accuracy and precision of the classical least square estimation or prediction ability can be improved by adding spectral shapes to either or both of the calibration step or the prediction step that describe the effects on the sample response from constituents that are present in the sample or parameters that affect the optical response of the sample but whose concentrations or values are not in the calibration data base.